Thermal Charm Production in Quark-Gluon Plasma at LHC

0.13 MB·

by  Ben-Wei Zhang

Thermal Charm Production inQuark-GluonPlasmaatLHC
Ben-Wei Zhang ,1 Che Ming Ko,1 and Wei Liu1
∗
1CyclotronInstituteandPhysicsDepartment, TexasA&MUniversity,CollegeStation,Texas77843-3366 USA
Charmproductionfromthequark-gluonplasmacreatedinthemidrapidityofcentralheavyioncollisionsat
theLargeHadronCollider(LHC)isstudiedinthenext-to-leadingorderinQCD.Usingaschematiclongitudi-
nallyboost-invariantandtransversallyexpandingfire-cylindermodel,wefindthatcharmproductioncouldbe
appreciablyenhancedatLHCasaresultofthehightemperaturethatisexpectedtobereachedintheproduced
quark-gluonplasma. Sensitivitiesofourresultstothenumberofcharmquarkpairsproducedfrominitialhard
scattering,theinitialthermalizationtimeandtemperatureofthequark-gluonplasma,andthecharmquarkmass
arealsostudied.
PACSnumbers:24.85.+p,25.75.-q,12.38.MH
8
0
0 I. INTRODUCTION ductionisfrominitialhardscatteringsofpartonsinthenucle-
2 onsfromthe collidingnuclei, which happensat a time scale
n In heavy-ion collisions at relativistic energies, hadrons of about 1/mc 0.15 fm/c with mc being the charm quark
∼
a composedofconfinedquarksandgluonsareexpectedtodis- mass. It contributes about 3 charm quark pairs in one unit
J solveintotheirconstituentsandformanextendedvolumeof of midrapidity in Au+Au collisions at √sNN =200 GeV at
1 RHIC, and this number is increased to about 20 in Pb+Pb
quark-gluon plasma (QGP). Experiments at the Relativistic
1 Heavy Ion Collider (RHIC) have indeed shown that the re- collisionsat √sNN =5.5 TeV at LHC [14, 15]. These num-
sultsare consistentwiththe formationof a stronglyinteract- bers have, however, substantial uncertainty, particularly in
]
h ing quark-gluon plasma during the initial stage of the colli- extrapolating to LHC energy from that at RHIC [12, 14],
-t sions [1, 2, 3, 4]. Many observables have been proposed to where measurements from PHENIX for p+p collisions [16]
l probethepropertiesofthequark-gluonplasma. Amongthem andSTARford+Aucollisions[17]disagreebyaboutafactor
c
of 2. The prethermal production is from produced partonic
u is the productionof particles consisting of heavy charm and
matter before it reaches thermalequilibrium. Based on con-
n bottomquarks.Inparticular,itwassuggestedthattheproduc-
[ tion of charmonium J/y in relativistic heavy ion collisions sideration of minijet partons, it has been shown in Ref.[18]

mightbesuppressedasaresultofitsdissociationinthepro- that this contribution is unimportant compared to that from
2
directproduction. Forthermalproduction,itcanbefromthe
v duced quark-gluon plasma due to the Debye screening [5].
thermalizedpartonicmatterorquark-gluonplasmaaswellas
4 Although recent studies based on the lattice QCD have in-
8 dicated that the J/y can survive in the quark-gluon plasma from the hadronic matter formed after the hadronization of
6 attemperaturesuptoabouttwicethedeconfinementtemper- the quark-gluon plasma. For thermal production of charm
1 ature [6, 7], J/y production in relativistic heavy ion colli- quarkpairsfromthequark-gluonplasma,studiessofarhave
. beenbasedonthelowest-orderQCDprocessofgluon-gluon
9 sionsmaystillbesuppressedinheavyioncollisionsatRHIC
0 [8,9,10]. However,iftheinitialproducedcharmquarkpairs fusion. The results show that its contribution depends sen-
7 are large such as in heavy ion collisions at LHC, J/y pro- sitively on the initial temperature of produced quark-gluon
0 plasma [19, 20, 21, 22, 23], and could be important if the
ductioncouldbeenhancedthroughregenerationfromcharm
v: andanticharmquarksinthequark-gluonplasma[11]aswell initial temperature is high. In hadronic matter, charmed
i as throughstatistical productionfrom [12] or coalescenceof hadron can be produced from reactions such as p N DL c
X [24], r N DL [25], andNN NL D [26]. Accor→dingto
[13]charmandanticharmquarksduringhadronizationofthe c c
ar quark-gluonplasma. These mechanismsfor J/y production Ref.[27] b→asedon theHadron-S→tringDynamics(HSD) with-
outapartonicstage,secondarymeson-baryonreactionsinthe
wouldbecomeevenmoreimportantifthereareothersources
hadronicmattergivesrisetoapproximately11%enhancement
forcharmquarkproductioninrelativisticheavyioncollisions.
ofcharmpairproductioninheavyioncollisionsatRHICen-
Tousecharmoniumandbottomoniumproductioninrelativis-
ergies. This contribution is, however, expected to be much
ticheavyioncollisionsasadiagnostictoolfortheproperties
smaller if onetakesinto accountthe formationof thequark-
ofproducedquark-gluonplasma[9,10,11,12],itisthuses-
gluon plasma in the collisions, which would lead to a final
sentialtounderstandtheproductionmechanismofcharmand
hadronicmatterwithsignificantlylowerdensityandtempera-
bottomquarksinthesecollisions.
turethanintheHSDmodel.
In heavy ion collisions, there are generally four different
contributionstocharmproduction:directandprethermalpro- ForAu+Aucollisionsat√sNN =200GeVatRHIC,charm
production is dominated by direct production as the contri-
ductionfrompartonicinteractionsaswellasthermalproduc-
bution from thermal production is negligible due to the rel-
tionfrompartonicandhadronicinteractions. Thedirectpro-
atively low initial temperature ( 350 MeV) of produced
∼
quark-gluon plasma. Since the thermal production rate in-
creasesexponentiallywithtemperatureaswewillshowlater,
thermalproductionofcharmquarkpairsinPb+Pbcollisions
OnleavefromInstituteofParticlePhysics,CentralChinaNormalUniver-
∗
sity,Wuhan430079,China at √sNN =5.5 TeV at LHC, in which a quark-gluonplasma
2
withmuchhigherinitialtemperatureanddensitythanthoseat gluon-quarkinteractions
RHICisexpectedtobecreated,willbegreatlyenhanced.On
theotherhand,charmproductionfrominitialhardscattering g+q c+c¯+q,
→
ofcollidingnucleonsincreasesonlylogarithmicallywith the g+q¯ c+c¯+q¯, (5)
collisionenergy. Inmoststudiesontheproductionofcharm →
particlesatLHC,ithasbeenassumed,however,thatthermal as their contributions are less important compared to other
productioncanbeneglectedaswell, andonlythedirectpro- processes, especially the gluon-gluon fusion process of
duction from initial hard scattering has been considered. In Eq.(4). This is largely due to the smaller color factor from
thepresentstudy,wewillconsiderthermalcharmproduction the gluon-quarkinteraction vertex as compared to that from
atLHCandcompareitscontributiontothatduetodirectpro- thegluon-gluonvertex,andthefactthatthecharmpaircanbe
ductiontoseewhetherthisassumptioncanbejustifiedorun- producedfrom two incoming gluon lines in gluon-gluonfu-
der what conditionsit can be considered as a good approxi- sion process, whereasin the gluon-quarkprocessit can only
mation. comefromasinglegluonline[30].
Charmproductioninaquark-gluonplasmahasbeenprevi-
ouslystudiedinthelowest-orderQCD[19,20,21,22,28,29].
Inourstudy,wewillconsiderthermalcharmproductioninthe 12
next-to-leadingorderandinvestigateitssensitivitytotheini- Next-to-leading order Next-to-leading order
tial conditions of the produced quark-gluonplasma at LHC. Leading order 8 Leading order
We find that based on a reasonable estimation of the initial
8
conditions,thermalcharmproductioncouldplayanimportant b) b)
roleindeterminingthetotalnumberofcharmpairsproduced
inrelativisticheavyioncollisions. m (qq m (gg4
This paper is organized as follows. In the next section, s 4 s
wediscusstheprocessesandcorrespondingcrosssectionsfor
thermalcharmproductionin theleadingorderaswellas the
next-to-leadingorderinQCD.InSectionIII,weintroducethe
0 0
time evolution of formed quark-gluon plasma in heavy ion 2 4 6 8 10 2 4 6 8 10
1/2 1/2
collisionsatLHCandderivetherateequationforcharmpro- s (GeV) s (GeV)
ductioninthequark-gluonplasma. InSectionIV,wepresent
the numerical results on thermal charm production at LHC FIG.1: (Color online) Charmquarkpair production crosssections
asfunctionsofcenter-of-massenergyfromquark-antiquarkannihi-
and their dependence on the initial conditions and other pa-
lation (left panel) and gluon-gluon fusion (right panel) in both the
rameters in the model. Finally, a summary and conclusions
leading(dashedline)andthenext-to-leading(solidline)order. The
aregiveninSectionV.
charmquarkmassistakentobemc=1.3GeV,whilelightquarks
andgluonsaremassless.
II. CHARMPRODUCTIONFROMQUARKANDGLUON
The total cross section for charm production in the next-
INTERACTIONS
to-leading-orderinQCD hasbeenpreviouslyderived. Inthe
presentstudy,weusetheresultsgiveninRefs. [30,31],where
A. charmproductioncrosssections
thesoftdivergencesinthenext-leading-order2 3processes
→
areeliminatedbycorrespondingdivergencesinvirtualcorrec-
Previous studies of thermal charm production in a quark- tionstotheleading-order2 2processes,whiletheultravi-
gluon plasma [19, 20, 21, 22, 23] were all based on the fol- olet divergencesin virtual c→orrections are handled by renor-
lowingleading-orderprocessesinQCD: malizationinthemodifiedminimal-subtraction(MS)scheme
[32, 33]. Choosing the QCD running coupling constant as
q+q¯ c+c¯, (1) a (m ) and the renormalization scale as the charm quark
→ s c
g+g c+c¯. (2) mass,wehavecalculatedthecrosssectionsfortheproduction
→
of charm pairs with charm quark mass m =1.3 GeV from
c
In the present study, we include also the next-leading-order masslesslightquarksandgluons,andtheyareshowninFig.1
processes as functions center-of-mass energy for the quark-antiquark
annihilation(leftpanel)andgluon-gluonfusion(rightpanel)
q+q¯ c+c¯+g, (3) processes.Thedashedandsolidlinescorrespondtoresultsfor
→
g+g c+c¯+g, (4) theleadingorderandthenext-to-leadingorder,respectively.It
→
isseenthatforcharmproductionfromquark-antiquarkanni-
and the interferences between the leading-order processes hilation,thenext-to-leadingordergivesalargercrosssection
withtheirvirtualcorrectionsduetovertexcorrectionsandself atlowenergiesbutasmalleroneathighenergies. Thecross
energy insertions as their contributions are of same order in section for charmproductionfromgluon-gluonfusion is, on
the QCD coupling as the next-leading-order processes. We theotherhand,significantlylargerinthenext-to-leadingorder
neglect, however, next-leading-orderprocesses involvingthe thanintheleadingorderatallenergies.
3
B. thermalaveragedcharmproductioncrosssections III. CHARMPRODUCTIONINHEAVYIONCOLLISIONS
ATLHC
Inthekineticmodeltobeusedinthenextsectionforstudy-
ingcharmquarkproductionandannihilationinaquark-gluon Usingabovethermalaveragedcharmproductioncrosssec-
plasma,thermalaveragedcrosssectionsareneeded. Interms tions,westudyinthissectionthetimeevolutionoftheabun-
ofthethermaldistributionfunctions f(p)ofquarksandglu- dance of charm quark pairs in heavy ion collisions at LHC
i
onsinthequark-gluonplasmaandtherelativevelocityv of using a kinetic model based on the rate equation that takes
ab
two initial interacting partons a and b, the thermal averaged intoaccountbothproductionandannihilationofcharmquarks
crosssections forthereactionab cdisgivenby[34] in the produced quark-gluon plasma. For the dynamics of
ab cd
→ → the quark-gluon plasma, we describe it by a schematic hy-
hs ab→cdvi = Rd3padd33ppbadfa3(ppbaf)af(bp(ap)bf)bs (apbb→)cdvab darreodinynthaemrmicamlaonddelchanemdiacsasluemqeuiltihbartiubmothduqruinagrktsheanevdoglulutioonns.
= [4a 2KR(a )a 2K (a )] 1 We furtherassume thatall producedcharmquarksincluding
a 2 a b 2 b − thoseproduceddirectlyfromtheinitialhardcollisionsarealso
¥
dz[z2 (a +a )2][z2 (a a )2] inthermalequilibrium,althoughnotinchemicalequilibrium.
× Zz0 − a b − a− b Thelatterisconsistentwithobservedlargeellipticflowofthe
K (z)s (s=z2T2), (6) electronsfromcharmedmesondecaysinheavyioncollisions
1
× at RHIC [37, 38], which requires that charm quarksinteract
with a i =mi/T, z0 =max(a a+a b,a c+a d), and K1 being stronglyinthequark-gluonplasmaandthusarelikelytoreach
themodifiedBesselfunction. thermalequilibrium[39,40,41].
2.5 6 A. therateequation
Next-to-leading order Next-to-leading order
Leading order 5 Leading order
2 The time evolution of the number density of charm quark
pairs n in an expanding quark-gluon plasma can be de-
b)1.5 b)4 scribedcbc¯ytherateequation[42,43]:
m> ( m> (3
vqq 1 vgg ¶ µ(ncc¯uµ) = Rqq¯ cc¯+Rqq¯ cc¯g+Rgg cc¯+Rgg cc¯g
→ → → →
s< s<2 R R R R . (8)
cc¯ qq¯ cc¯g qq¯ cc¯ gg cc¯g gg
− → − → − → − →
0.5
1 In the above, uµ =g (1,v) is the four velocity of a fluid ele-
mentin the quark-gluonplasma with velocityv and the cor-
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 responding Lorentz factor g , and the terms on the left hand
T(GeV) T(GeV)
sideofaboveequationarethecharmpairproductionandan-
nihilationrates. To the next-to-leadingorder,the charmpair
FIG. 2: (Color online) Thermal averaged cross sections for charm
productionrateisgivenby
pair production from quark-antiquark annihilation (left panel) and
gluon-gluonfusioninaquark-gluonplasmaasfunctionsoftemper- R = s v n n ,
ature. The charm quark mass is taken to be mc =1.3 GeV while qq¯→cc¯ h qq¯→cc¯ i q q¯
quarksandgluonsaretakentohavethermalmassesgivenbyEq.(7). R = 1 s v n2,
gg→cc¯ 2h gg→cc¯ i g
R = s v n n ,
Inthequark-gluonplasma,quarksandgluonsacquirether- qq¯ cc¯g qq¯ cc¯g q q¯
→ h → i
malmasses. Asanexploratorystudy,weincludethiseffectin 1
R = s v n2, (9)
their distributionfunctionsbut notin the calculation of their gg→cc¯g 2h gg→cc¯g i g
scattering cross sections. For the thermal masses of quarks
where n , n and n denote the gluon, quark, and antiquark
andgluons,theyaretakentobe[35,36]: g q q¯
densities in the quark-gluon plasma, respectively, and they
m =gT/√6 and m =gT/√2, (7) aretakentohavetheirequilibriumvalues. Theleading-order
q g crosssectionss ands inaboveequationarecom-
qq¯ cc¯ gg¯ cc¯
where g is the QCD coupling constant and is taken to have puted from the pr→ocesses in E→q.(1), while the next-leading-
tmhe=va1lu.3eGge=V, the4pathesr(m2paTla)v.erWagiethdcthroesschsaercmtionqufaorrkchmaarsms porrodceerscsreossisnsEeqc.t(i3o)nsansdqtq¯h→eccv¯girtaunadlcs ogrg¯r→ecctc¯igoninsctloudtheebleoathdinthge-
c
production in apquark-gluon plasma is shown in Fig. 2 as orderprocesses.
functions of the temperature of the quark-gluon plasma for
quark-antiquarkannihilation(leftpanel)andgluon-gluonfu-
sion(rightpanel). Forbothreactions,thermalaveragedcross
sections are significantly larger in the next-to-leading order InFig.3, we showthethermalcharmproductionrateasa
(solidline)thanintheleadingorder(dashedline). functionoftemperaturewithmassivequarksandgluonsina
4
1 n 2
1.0 Next-to-leading order R = s v (neq)2 cc¯ . (11)
Leading order cc¯g→gg 2h gg→cc¯g i g (cid:18)neccq¯(cid:19)
0.1 In the above, we have assumed that the density of quarks is
eq eq
thesameasthatofantiquarks,i.e.,n =n .
) -2 q q¯
4 10 With above equations and taking light quarks and gluons
m
5.0 to be in thermal and chemical equilibrium, the rate equation
f
c/10-3 4.0 Eq.(8)canberewrittenas
( o
R ati3.0 ¶ (n uµ)= ( s v + s v )(neq)2
10-4 R2.0 µ cc¯ h qq¯→cc¯ i h qq¯→cc¯g i q
10-5 1.00.2 0.T4 (GeV0).6 0.8 + 12(hs gg→cc¯(cid:2)vi+hs gg→cc¯gvi)(negq)2(cid:21)"1−(cid:18)nneccq(cid:19)2#.
-6 (12)
10
0.2 0.3 0.4 0.5 0.6 0.7 0.8
T (GeV)
B. collisiondynamicsatLHC
FIG. 3: (Color online) Thermal charm quark production rate as a
functionoftemperatureformassivepartonsandcharmquarkmass In central heavy-ion collisions at RHIC, the particle dis-
mc=1.3GeV.Theinsetgivestheratioofthecharmproductionrate tribution at mid-rapidity is approximately uniform, so the
inthenext-to-leadingordertothatintheleadingorder. Bjorkenboost-invarianthydrodynamicsolutionisagoodap-
proximationfordescribingthetimeevolutionofthehotdense
matter formedin these collisions[44]. Such a modelindeed
thermallyandchemicallyequilibratedquark-gluonplasma. It gives a good description of the experimental data at RHIC
isseenthatthecharmproductionrateincreasesalmostexpo- [45]. Also, recentstudiesbased on the AdS/CFT correspon-
nentiallywithincreasingtemperature.Withtheinitialtemper- dencehavedemonstratedthatatlatepropertimesthesystem
atureT0 350MeVincentralheavyioncollisionsatRHIC, produced in relativistic heavy-ion collisions in the strongly-
≃
thethermalcharmproductionrateismorethantwoordersof coupled regime described by gauge-gravity duality exhibits
magnitudesmallerthanthatatthetemperatureT0 700MeV an energy density scaling with characteristics similar to the
≃
expectedtobereachedincentralheavyioncollisionsatLHC. Bjorkenhydrodynamicsolution[46,47]. Wethusexpectthat
Thisthusjustifiestheneglectofthermalcharmproductionat the longitudinaldynamics in heavy ion collisions at LHC is
RHIC. The inset in Fig. 3 gives the ratio of the charm pro- alsoboostinvariant. Toincludetheeffectoftransverseflow,
ductionrateinthenext-to-leadingordertothatintheleading wefurtherincludeanacceleratedtransverseexpansion,result-
order,whichisseentovaryfrom 4.5atlowtemperaturesto inginacylindricalsymmetryinthegeometryofthecollision.
∼
1.8athightemperatures. Intermsofthecylindricalcoordinatesr andj aswellasthe
∼
Since the charm pair annihilationrate and productionrate propertimet andthespace-timerapidityh definedby[42,43]
become equal when the charm quark abundance reaches
1 t+z
chemicalequilibrium,i.e, t = t2 z2, h = ln , (13)
− 2 t z
−
Reccq¯→qq¯ = Reqqq¯→cc¯=h1s qq¯→cc¯vineqqneq¯q, wdeensthiteyndhisatvriebuuthio=npiunj t=he0t.ranIfsvweersfeuprtlhanereaasnsdumtaekeathueniafovremr-
Reccq¯→gg = Regqg→cc¯= 2hs gg→cc¯vi(negq)2, age over the radial coordinate, the left hand side of the rate
Req = Req = s v neqneq, equationEq.(12)canthenbeexpressedas:
Rcecccq¯¯gg→→qggq¯ = Rqegqgq¯→→cccc¯¯gg=h12hqsq¯g→g→cc¯gcc¯giviq(negq¯q)2, (10) t R21(t )t¶¶ (t R2(t )ncc¯hut i), (14)
whereneqistheequilibriumdensitywhileReqistherateeval- where R(t ) denotes the transverse radius of the system, and
ut istheaveragedt componentofthefourvelocitydefined
uatedwiththeequilibriumdensities,thecharmpairannihila-
h i
as
tionratesintheleadingorderandthenext-leadingordercan
thusbewrittenas 2 R(t )
t t
u = drru (r). (15)
n 2 h i R2(t )Z0
Rcc¯→qq¯ = hs qq¯→cc¯vi(neqq)2 necqc¯ , Atmid-rapidity,thefourvelocityofafluidelementuµ can
(cid:18) cc¯(cid:19) be described by two independent boosts in longitudinal and
R = 1 s v (neq)2 ncc¯ 2, radial directions [48], with the radial flow velocity b r given
cc¯→gg 2h gg→cc¯ i g neq by
(cid:18) cc¯(cid:19)
Rcc¯g→qq¯ = hs qq¯→cc¯gvi(neqq)2(cid:18)nnecccqc¯¯(cid:19)2, ut =g r= 11−b 2r . (16)
p
5
Assuming the usual ansatz for the radial expansion velocity time t 0.07 fm/c. The glasma then evolves into a ther-
[42,43,48,49] malized∼quark-gluonplasma at a proper time t . Assuming
0
that the energy density decreases inversely with the proper
b (t ,r)= dR r , (17) time during this stage, its value at t 0 0.2 fm/c is then
r dt R e 245 GeV/fm3, corresponding to an∼initial temperature
0
(cid:16) (cid:17) ≈
T 633 MeV. Since the result from the Color Glass Con-
wehave 0
≈
densatemodeldependsonthe fourthpowerofthe saturation
1 1 momentum,thepredictedinitialtemperaturethushasalarge
t
u = dy . (18)
h i Z0 1 (dR/dt )2y uncertainty.ThesameistruefortheAMPTmodelasthepre-
− dictedtotaltransverseenergyissensitivetothepartonstruc-
Withthetimeevolutionopfthetransverseradiusofthefire- turefunctionofthenucleon,theinitialnuclearshadowing,and
cylindertakentobe[50] thesaturationofproducedgluons.Therefore,wewillconsider
in thefollowingalso the scenariowithan initialenergyden-
R(t )=R +a(t t )2/2, (19) sity whichis50%ora factoroftwo largerlargerthanabove
0 0
−
value,correspondingtoaninitialtemperatureofabout700or
where R0 andt 0 are the initialradiusandpropertime of the 750MeV,respectively.
fire-cylinder,respectively,andadenotesthetransverseaccel- WenoteforcentralAu+Aucollisionsat√sNN =200GeV
eration,thevolumeofthefireballatpropertimet isthengiven at RHIC, the total transverse energy from the AMPT model
by is about 1000 GeV. Taking a proper time t =0.5 fm/c for
0
the formation of an equilibrated quark-gluon plasma as re-
V(t )=p R2(t )t . (20) quired by the hydrodynamic model to explain the observed
hadron elliptic flows [53, 54, 55], the initial energy density
Inthefollowing,wechoosetheinitialtransverseradiusofthe
and temperature of produced quark-gluon plasma are then
fire-cylindertobetheradiusofthecollidingPbnucleus,i.e., e 33 GeV/fm3 and T 383 MeV, respectively. These
R0 =7.0 fm, and a transverse acceleration a=0.1 c2/fm in v0al≈uesareagainsimilarto0th≈osefromtheColorGlassConden-
ordertoobtainareasonablefinaltransverseflowvelocity.
satemodel,whichgivesanenergydensitye 130GeV/fm3
≈
at t = 0.1 fm/c, leading thus to e 26 GeV/fm3 and
0 0
T 361 MeV at t = 0.5 fm/c. Wit≈h such a low initial
C. timeevolutionofthetemperatureofthequark-gluon 0 0
≈
plasma temperature,charmproductionfromthe quark-gluonplasma
is thusnegligibleaccordingto the productionratesshown in
Fig.3.
Toestimatetheinitialtemperatureofproducedquark-gluon
plasma, we use two differentmodels: A Multi-phase Trans-
port (AMPT) model [51] and the model based on the Color
Glass Condensate [52]. Using the AMPT model, we obtain 0.8
aninitialtransverseenergyofabout3000GeVperunitrapid- T = 750 MeV
0
ity at the midrapidityin centralPb+Pb collisions at √sNN = T = 700 MeV
0
5.5TeV. Mostof thisinitialenergyisconcentratedwithina
T = 630 MeV
transverseradiusofabout4.7fmthatissmallerthantheradius 0.6 0
ofthecollidingnucleiasaresultoftheirsurfacediffuseness. )
V
Ifwetakeanearlierinitialpropertimeoft 0=0.2fm/cthan e
G
that at RHIC, the initial energy density of produced quark-
(0.4
gluonplasmaisthenabout T
dE /dy 3000
e T 226GeV/fm3. (21)
0≈ p R20t 0 ≈ p ×4.72×0.2 ≈ 0.2
Usingtherelation
0 1 2 3 4 5 6 7
t
37p 2T4 (fm/c)
e = (T/160MeV)4 GeV/fm3 (22)
30 ≈
FIG. 4: (Color online) Time evolution of the temperature of the
for the energy density of a quark-gluon plasma with mass- quark-gluonplasmaformedinPb+Pbcollisionsat√sNN=5.5TeV
less quarks and gluons, the initial temperature of the quark- for different initial temperatures and same initial proper time t 0=
gluon plasma formed at LHC is thus T 620 MeV. This 0.2fm/c.
0
≈
number turns out to be very close to that predicted by the
Color Glass Condensate model. As shown in Ref.[52], the For the time evolution of the temperature of produced
glasma formed after Pb+Pb collisions at √sNN = 5.5 TeV quark-gluonplasmaatLHC,wedetermineitusingtheentropy
hasanenergydensityofaboute 700GeV/fm3 ataproper conservation,andtheresultsareshowninFig.4forthethree
∼
6
initialtemperaturesT =630,700,and750MeV.Inallcases, MeV when the proper time is t 6.4 fm/c, the number of
0 C
≈
thecriticaltemperatureforthequark-gluonplasmaphasetran- charm pairs is about 27 in the next-to-leading order (solid
sitiontothehadronicmatteristakentobeT =170MeV. It line), which is about 30% larger than the number due to di-
C
is seen that the temperature decreases very quickly with the rect productionfrom initial hard collisions. The final charm
propertimeforallthreeinitialtemperatures,droppingbelow pairnumberisreducedtoabout22ifonlythermalproduction
400MeVafter about0.5, 1, and1.5fm/c forT =630,700, from leading-ordercontribution is included as shown by the
0
and750MeV,respectively. dashedline.
We note that the final charm quark pair number is much
largerthanitschemicallyequilibriumvalueatT =170MeV,
C
IV. RESULTS whichisabout4.7.Thisisnotsurprisingasthetimeforcharm
eq
pairs to reach their equilibrium number n at certain tem-
c
For central Pb+Pb collisions at √sNN =5.5 TeV at LHC, perature,givenbyt eq =neccq¯/2Rcc¯ with Rcc¯ denotingthepro-
thenumberofcharmpairsproducedinoneunitrapidityfrom duction rate at that temperature, increases dramatically with
initial hard nucleon-nucleoncollisions is about 20 at midra- decreasing quark-gluon plasma temperature from a value of
pidity according to the next-to-leadingorder pQCD calcula- a few fm/c at T =700 MeV to a few thousands of fm/c at
tions[12]. Neglectingthecontributionfrompre-thermalpro- T = 170 MeV. It is thus not possible for charm quarks to
c
duction,wesolvetherateequationwiththisinitialnumberof reach chemical equilibrium during the finite lifetime of the
charmquarkspairsandcertaininitialtemperatureandproper producedquark-gluonplasmaatLHC.Sincetheinitialnum-
time forthe producedquark-gluonplasma. We willvarythe berofcharmquarkpairsproducedfromhardscatteringofini-
initial temperature and proper time to study their effects on tialnucleonsislargeatLHCandthecharmproductionrateis
thefinalnumberofcharmquarkpairsproducedinthesecolli- also larger during the early stage of the quark-gluonplasma
sions. when its temperature is high, there are more charm pairs in
the quark-gluon plasma than the equilibrium number at the
criticaltemperature. Asthequark-gluonplasmaexpandsand
cools, the charm annihilation rate decreases, making it less
likely to destroy the produced charm pairs and leading thus
to an over-saturation of the charm abundance at the critical
30 temperature.
c 50
c
N
20 T=700 MeV, t =0.2 fm/c
m0c=1.3 GeV, m0assive partons 40
Next-to-leading order
Leading order
30
c
c
N
10
0 1 2 3 4 5 6 20
t (fm/c)
N (t ) = 40
cc 0
FIG.5:(Coloronline)Numberofcharmpairsasafunctionofproper 10 mT0==710.30 GMeeVV, ,m t0a=ss0i.v2e f pma/rct o ns Ncc(t 0) = 20
timeincentralPb+Pbcollisionsat√sNN =5.5TeVintheleading c N (t ) = 10
cc 0
order(dashedline)andthenext-to-leadingorder(solidline)inQCD.
0
0 1 2 3 4 5 6
t (fm/c)
We first consider the case of an initial temperature T =
0
700MeVandpropertimet =0.2fm/c.Forthecharmquark
0
mass,wetakeittobem =1.3GeVasthatextractedfromthe FIG.6:(Coloronline)Numberofcharmpairsasafunctionofproper
c
experimentaldataine+e and ppcollisions. Totakeintoac- time in central Pb+Pb collisions at √sNN =5.5 TeV for different
−
count the medium effect on quarks and gluons, we use the numberofcharmquarkpairsproducedfrominitialhardscattering.
thermalmassesgiveninEq.(7). Thetimeevolutionoftheto-
tal number of charm quark pairs obtained from these initial Since there is substantial uncertainty in the charm quark
conditionsin centralPb+Pbcollisionsat√sNN =5.5TeV is pairsproducedfrominitialdirectproduction[12,14],wehave
showninFig. 5. Itisseen thatincludingthermalproduction also studied thermal charm productionusing differentinitial
enhances the total number of final charm pairs as compared charm quark pair numbers. Varying the initial charm pair
to that from initial direct production. The charm pair num- number by a factor of two, we have repeated above calcu-
ber reaches the peak value at t 2 fm/c and then deceases lations, and the results based on the next-to-leadingorder in
∼
with the proper time. At the critical temperature T =170 QCDareshowninFig.6. Forinitialnumbersofcharmquark
C
7
pairsof10,20and40,thefinalnumbersarefoundtobeabout 500MeV. Resultsforthisinitialtemperatureaswellasthose
19, 27, and 45, respectively. Thermal production of charm forthetemperaturesof560,and600MeV,whichcorrespond,
quarksfromthe quark-gluonplasma thusbecomesmoreim- respectively, to an increase of the initial energyby 50% and
portantastheinitialcharmpairnumberbecomessmaller. 100%ofthatforT =500MeV,showthatthefinaltotalnum-
0
Returning to the case of 20 initial charm quark pairs, we berofcharmquarkpairsisonlyslightlyreducedcomparedto
notethatinabovecalculations,wehaveusedmassivequarks thatforaninitialpropertimet 0=0.2fm/c,i.e.,21,26,and
and gluons given by the thermal QCD calculations. To see 32forabovethreetemperatures.
how these masses affect thermal charm production, we have Thecharmquarkmassalsoaffectsthecontributionofther-
carriedoutsimilarcalculationsasabovewithmasslessquarks mal charm production. With mc =1.5 GeV and the initial
andgluonsbutsameinitialconditionsofT0=700MeVand proper time t 0 =0.2 fm/c, the final total number of charm
t = 0.2 fm/c as well as the charm quark mass of m = quarkpairsisreducedtoabout20,22and25forthethreeini-
0 c
1.3 GeV. We find that the resulting numberof charm quark tialtemperaturesT0=630,700,and750MeV,respectively.
pairsatT =170MeVdiffershardlyfromthemassivecase.
C
Thereasonforthisisthatalthoughusingmasslessquarksand
gluonsreducesthe thermalaveragedcharmproductioncross V. SUMMARYANDDISCUSSIONS
sections,theeffectislargelycompensatedbythelargerquark
andgluondensitiesiftheyaremassless. In this paper, we have carried out the first calculation of
thermal charm production at next-to-leading-order in QCD.
ModelingcentralheavyioncollisionsatLHCbyaschematic
40 longitudinallyboostinvariantandtransverselyexpandingfire-
cylinderofquark-gluonplasma, wehaveevaluatedthenum-
ber of charm quark pairs producedin these collisions. With
aninitialtemperatureof700MeVforanequilibratedquark-
30 gluonplasmaataninitialpropertimeof0.2fm/candacharm
quarkmassof1.3GeV,wehaveobtainedabout30%enhance-
ment in the production of charm quarks than that produced
c
c directly from initial hard collisions, if the latter is taken to
N
20 be 20 pairs at midrapidity according to the next-to-leading
t =0.2 fm/c, m=1.3 GeV orderpQCD calculations. Aboutequalcontributionsare ob-
0 c
tainedfromtheleadingorderandthenext-leadingorderpro-
T = 750 MeV
0 cesses. This result is, however, sensitive to the initial con-
10 T = 700 MeV
0 ditions for the produced quark-gluon plasma as well as the
T = 630 MeV
0 charm quark mass. The enhancement is increased to about
0 1 2 3 4 5 6 80%iftheinitialtemperatureisincreasedto750MeVbutre-
t (fm/c) ducedto about10%if theinitialtemperatureis decreasedto
630 MeV. Delaying the proper time at which a thermalized
FIG.7: (Coloronline)Totalnumberofcharmpairsasafunctionof quark-gluonplasmaisformeddoesnotaffect,however,much
propertimeincentralPb+Pbcollisionsat√sNN =5.5TeVfordif- thermalcharmquarkproductionastheeffectduetodecreased
ferentinitialtemperaturesofthequark-gluonplasmabutsameinitial initialtemperatureiscompensatedbythatfromtheincreased
propertimet 0=0.2fm/c. volumeofthequark-gluonplasma.Changingthecharmquark
masshas, ontheotherhand,alargeeffectonthermalcharm
As previously mentioned, the charm production rate ex- quarkproductionfromthequark-gluonplasma. Withalarger
hibits an exponential increase with the temperature of the charmquarkmassofmc=1.5GeV,thermalcharmquarkpro-
quark-gluonplasma. The total numberof charm quarkpairs duction from the quark-gluon plasma becomes unimportant
produced in heavy ion collisions thus depends on the initial evenforaninitialtemperatureofT0=700MeV.Finally,ther-
temperatureoftheexpandingquark-gluonplasma. InFig.7, malcharmproductionfromthequark-gluonplasmabecomes
we show the total number of charm pairs as a function of moreimportantifthenumberofdirectlyproducedcharmpairs
proper time in central Pb+Pb collisions at √sNN =5.5 TeV frominitialhardscatteringissmall.Theseresultsarenotonly
for different initial temperatures of the quark-gluon plasma ofinterestintheirownrightbutalsousefulforunderstanding
but same initial proper time t =0.2 fm/c. It is seen that charmoniumproductioninrelativisticheavyioncollisionsas
0
thefinalnumberofcharmquarkpairsatthecriticaltempera- itsproductionfromthequark-gluonplasmaisproportionalto
tureT =170MeVcalculatedinthenext-to-leading-orderde- thesquareofthecharmquarknumbers.
C
creasestoabout22forT =630MeVandincreasestoabout
0
33forT =750MeV.
0
Wehavealsoconsideredtheeffectofinitialpropertimeon Acknowledgments
thermal charm productionby using t =0.5 fm/c as that in
0
heavy ion collisions at RHIC. The predicted initial temper- ThisworkwassupportedinpartbytheUSNationalScience
ature from the Color Glass Condensate model is then T = Foundation under Grant No. PHY-0457265 and the Welch
0
8
Foundation under Grant No. A-1358. Ben-wei Zhang was ofChinaunderprojectNo. 10405011andbyMOEofChina
furthersupportedbytheNationalNaturalScienceFoundation underprojectIRT0624.
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